A computational approach to Brauer Witt theorem using Shoda pair theory

TL;DR

This paper introduces a computational method leveraging Shoda pair theory to explicitly determine the cyclotomic algebras associated with the simple components of rational group algebras, addressing a longstanding problem in algebra.

Contribution

It provides a novel, efficient computational approach to explicitly describe the cyclotomic algebras in the Brauer-Witt theorem using Shoda pair theory.

Findings

Efficient algorithm for computing cyclotomic algebras

Explicit descriptions of simple components of QG

Enhanced understanding of central simple algebras

Abstract

A classical theorem due to Brauer and Witt implies that every simple component of the rational group algebra QG of a finite group G is Brauer equivalent to a cyclotomic algebra containing Q in its centre. The precise description of this cyclotomic algebra is not available from the proof of the Brauer-Witt theorem and it has been a problem of interest to determine the same in view of its central role in the study of central simple algebras. In this paper, an approach using Shoda pair theory is described, which is quite efficient from computational perspective.